Numerical solutions of reaction-diffusion equations: Application to neural and cardiac models

Ji, Yanyan Claire; Fenton, Flavio H.

doi:10.1119/1.4953167

Cited by 8 publications

(3 citation statements)

References 42 publications

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“…5 ) with time step and spatial step . Such discretization satisfies the criterion of stability for the difference schemes , where D is the diffusion coefficient 16 , 53 . To implement no-flux boundary conditions for an arbitrary cell shape, we applied the modified five-point stencil of the 2D Laplace operator , so that: where is the concentration of , or in position at time , is the binary mask representing the cell ( ) and the background ( ), is the diffusion coefficient of or (while the diffusion of the component is negligible) and is the reaction term (see Eqs.…”

Section: Methodsmentioning

confidence: 99%

Spatiotemporal development of coexisting wave domains of Rho activity in the cell cortex

Hladyshau

Kho

Nie

et al. 2021

Sci Rep

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The Rho family GTPases are molecular switches that regulate cytoskeletal dynamics and cell movement through a complex spatiotemporal organization of their activity. In Patiria miniata (starfish) oocytes under in vitro experimental conditions (with overexpressed Ect2, induced expression of Δ90 cyclin B, and roscovitine treatment), such activity generates multiple co-existing regions of coherent propagation of actin waves. Here we use computational modeling to investigate the development and properties of such wave domains. The model reveals that the formation of wave domains requires a balance between the activation and inhibition in the Rho signaling motif. Intriguingly, the development of the wave domains is preceded by a stage of low-activity quasi-static patterns, which may not be readily observed in experiments. Spatiotemporal patterns of this stage and the different paths of their destabilization define the behavior of the system in the later high-activity (observable) stage. Accounting for a strong intrinsic noise allowed us to achieve good quantitative agreement between simulated dynamics in different parameter regimes of the model and different wave dynamics in Patiria miniata and wild type Xenopus laevis (frog) data. For quantitative comparison of simulated and experimental results, we developed an automated method of wave domain detection, which revealed a sharp reversal in the process of pattern formation in starfish oocytes. Overall, our findings provide an insight into spatiotemporal regulation of complex and diverse but still computationally reproducible cell-level actin dynamics.

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Section: Methodsmentioning

confidence: 99%

Spatiotemporal development of coexisting wave domains of Rho activity in the cell cortex

Hladyshau

Kho

Nie

et al. 2021

Sci Rep

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“…(1) and (2) numerically using the forward time centered space method. 20,21 The fields are discretized on a 128 × 128 grid, such that x = j∆, y = i∆, t = nδt and i, j, n ∈ N. Differential operators are replaced by forward and central finite differences with spacing ∆ = 1 (length units) and time step δt = 0.25 (time units), as illustrated in Fig. 1(c).…”

Section: The Gray-scott Modelmentioning

confidence: 99%

Exploring complex pattern formation with convolutional neural networks

Scholz,

Scholz

2021

Preprint

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Many nonequilibrium systems, such as biochemical reactions and socioeconomic interactions, can be described by reaction-diffusion equations that demonstrate a wide variety of complex spatiotemporal patterns. The diversity of the morphology of these patterns makes it difficult to classify them quantitatively and they are often described visually. Hence, searching through a large parameter space for patterns is a tedious manual task. We discuss how convolutional neural networks can be used to scan the parameter space, investigate existing patterns in more detail, and aid in finding new groups of patterns. As an example, we consider the Gray-Scott model for which training data is easy to obtain. Due to the popularity of machine learning in many scientific fields, well maintained open source toolkits are available that make it easy to implement the methods we discuss in advanced undergraduate and graduate computational physics projects.

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“…Mainly, semi-linear reaction-diffusion PDEs are commonly used to model a variety of real-world phenomenon such as population dynamics and chemical reactions etc., [31], [40]. Many researchers have focused their attention on reactiondiffusion equations due to their wide range of applications, see [1], [11], [12], [17], [25], [33]. Random noise in dynamical systems is caused by external disruptions, measurement errors, and a lack of knowledge of specific parameters.…”

Section: Introductionmentioning

confidence: 99%

Exponential Stability Analysis of Stochastic Semi-Linear Systems With Lèvy Noise

et al. 2022

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The exponential stability of semi-linear stochastic partial differential equations (SPDEs) involving Lèvy type noise is investigated in this paper. By constructing an appropriate Lyapunov function, a new set of sufficient conditions are established in terms of linear matrix inequalities (LMIs) which ensures the mean-square exponentially stability (MSES) of given system with Neumann boundary conditions. Then the H ∞ performance index is introduced to eliminate the disturbance which occurs in the considered system. The boundary control gain is obtained by solving the LMI conditions using the standard MATLAB software. Finally, a numerical example is provided to demonstrate the usefulness of the proposed methods.

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Numerical solutions of reaction-diffusion equations: Application to neural and cardiac models

Cited by 8 publications

References 42 publications

Spatiotemporal development of coexisting wave domains of Rho activity in the cell cortex

Spatiotemporal development of coexisting wave domains of Rho activity in the cell cortex

Exploring complex pattern formation with convolutional neural networks

Exponential Stability Analysis of Stochastic Semi-Linear Systems With Lèvy Noise

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